Yes. The alternate ending consisted of footage not seen by many other than in Japan but it does contain a few shots from Godzilla's death as seen in the final cut. Godzilla still dies, but in the alternate ending Destroyer actually gets up after being shot down by the JSDF and the Super-X III, but is badly wounded and unable to fight. Godzilla in his rage grabs him by the horn and continues to pummel him with his free fist. While doing so he finally reaches the meltdown temperature of 1200 and the army rains freezer weapons on both of them. Destroyer ,due to being so close to Godzilla during his meltdown and the extreme cold created by the Super X III, dies right before Godzilla does as well. In the end, Junior still is revived so in essence it's more or less the same, but director Takao Owara decided to give the Monster King his much needed center stage as he died, and so new shots were added and Destroyah dies long before so. The Sony Tristar DVD contains no special features, what so ever, but the Toho DVD-r copy (which is available on Amazon) has this ending along with plenty of other features.

Yes, in fact there are three. keep in note these scenes can not be seen on the internet anywhere (as far as I have searched) or on the Sony Tristar DVD release of the film. In order to see them you must buy the actual Toho DVD, available on Amazon.com and monsterislandtoys.com.

(1) One has Junior entering Tokyo in the same style as he did in the final cut, along with the flying form of Destroyer ramming him, but it is shot from entirely different angles.

(2) The second is the highly talked about scene where Destroyer's chest opens up and fires an enormous blast at Godzilla, knocking him into a building and hurting him badly. Godzilla however recovers and counters with his breath weapon. While as the scene is pretty much complete, with sound effects and everything, it was apparently too early in production to add either Godzilla or Destroyer's beam weapons into the shots.

Yes, kind of. Unlike most movies that have an black backdrop during the credits, the Heisei Godzilla films had special footage that played during the credits. And because this was (at the time) to be Godzilla's final film, there are scenes from all seven Heisei Godzilla films, along with scenes from the original 1954 film "Gojira", inorder to commemorate the Monster king's death.